Unformatted text preview: 10/6/2021 Ch. 2 Key Equations - University Physics Volume 1 | OpenStax Key Equations
Multiplication by
a scalar (vector
equation) ⃗ ⃗ = Multiplication by
a scalar (scalar
equation for
magnitudes) = || Resultant of two
vectors ⃗ ⃗ ⃗ = + Commutative
law ⃗ ⃗ ⃗ ⃗ + = + Associative law ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ( + ) + = + ( + ) Distributive law ⃗ ⃗ ⃗ 1 + 2 = ( 1 + 2 ) The component
form of a vector
in two
dimensions ⃗ = ˆ + ˆ Scalar
components of
a vector in two
dimensions = − { = − Magnitude of a
vector in a plane 2
2
‾‾‾‾‾‾‾
‾ = √ + The direction
angle of a vector = tan −1 ( )
1/4 10/6/2021 Ch. 2 Key Equations - University Physics Volume 1 | OpenStax in a plane
Scalar
components of
a vector in a
plane
Polar
coordinates in a
plane = cos { = sin = cos { = sin The component
form of a vector
in three
dimensions ⃗ = ˆ + ˆ + ˆ The scalar zcomponent of a
vector in three
dimensions = − Magnitude of a
vector in three
dimensions 2
2
2
‾‾‾‾‾‾‾‾‾‾‾
‾ = √‾ + + Distributive
property ⃗ ⃗ ⃗ ⃗ ( + ) = + Antiparallel
vector to ⃗ Equal vectors ⃗ ˆ
− = − ˆ − − ˆ ⃗ ⃗ = ⇔ ⎧ = ⎪
⎨ = ⎪
⎩ = Components of
2/4 10/6/2021 Ch. 2 Key Equations - University Physics Volume 1 | OpenStax the resultant of
N vectors ⎧ ⎪ = 1 + 2 + … + =
∑
⎪
=1 ⎪ ⎪
⎨ =
⎪ ∑ = 1 + 2 + … + =1 ⎪ ⎪
⎪ =
⎩ ∑ = 1 + 2 + … + =1 General unit
vector ˆ = Definition of the
scalar product ⃗ ⃗ · = cos Commutative
property of the
scalar product ⃗ ⃗ ⃗ ⃗ · = · Distributive
property of the
scalar product ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ · ( + ) = · + · Scalar product
in terms of
scalar
components of
vectors ⃗ ⃗ · = + + Cosine of the
angle between
two vectors
Dot products of
unit vectors ⃗ cos = ⃗ ⃗ · ˆ · ˆ = ˆ · ˆ = ˆ · ˆ = 0 3/4 10/6/2021 Ch. 2 Key Equations - University Physics Volume 1 | OpenStax Magnitude of
the vector
product
(definition) ∣ ⃗ ⃗ ∣
∣ × ∣ = sin ∣
∣ Anticommutative
property of the
vector product ⃗ ⃗ ⃗ ⃗ × = − × Distributive
property of the
vector product ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ × ( + ) = × + × Cross products
of unit vectors ⎧ˆ
ˆ × = +ˆ
,
⎪
ˆ
⎨ˆ × ˆ = + ,
⎪
ˆ
ˆ
⎩ˆ × = +. The cross
product in terms
of scalar components of
vectors ⃗ ⃗ × = ( − )ˆ + ( − )ˆ 4/4 ...
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